Structural equation modeling
Structural equation modeling (SEM) is a general term used to describe either of two computer based statistical fit software packages: PLS-PA and LISREL/AMOS software. The methods were developed as alternatives to the more widely used systems of regression equation approached used in econometrics.
SEM evolved in three different streams: (1) systems of equation regression methods developed mainly at the Cowles Commission; (2) iterative maximum likelihood algorithms for path analysis developed mainly at the University of Uppsala by Karl Gustav Jöreskog; and (3) iterative canonical correlation fit algorithms for path analysis also developed at the University of Uppsala by Hermann Wold. Much of this development occurred at a time that automated computing was offering substantial upgrades over the existing calculator and analogue computing methods available, themselves products of the proliferation of office equipment innovations in the late 19th century.
Loose and confusing terminology has obscured exactly what SEM is doing with data. In particular, PLS-PA (the Lohmoller algorithm) is quite commonly confused with partial least squares regression, which is typically just called PLS. PLS regression tends to be useful with very large, multicolinear datasets, and finds applications in spectroscopy. PLS-PA, in contrast, is typically promoted as a method that works with small datasets when other estimation approaches fail; though this contention, even in the 1970s, was known not to be true; e.g. see (Dhrymes, 1972, 1974; Dhrymes & Erlat, 1972; Dhrymes et al., 1972; Gupta, 1969; Sobel, 1982)
Both LISREL and PLS-PA were conceived as iterative computer algorithms, with an emphasis from the start on creating an accessible graphical and data entry interface and extension of Wright’s (1921) path analysis. Early Cowles’ Commission work on simultaneous equations estimation centered on Koopman and Hood’s (1953) algorithms from the economics of transportation and optimal routing, with maximum likelihood estimation, and closed form algebraic calculations, as iterative solution search techniques were limited in the days before computers. Anderson and Rubin (1949, 1950) developed the limited information maximum likelihood estimator for the parameters of a single structural equation, which indirectly included the two-stage least squares estimator and its asymptotic distribution (Anderson, 2005) and Farebrother (1999). Two-stage least squares was originally proposed as a method of estimating the parameters of a single structural equation in a system of linear simultaneous equations, being introduced by Theil (1953a, 1953b, 1961) and more or less independently by Basmann (1957) and Sargan (1958). Anderson’s limited information maximum likelihood estimation was eventually implemented in a computer search algorithm, where it competed with other iterative SEM algorithms. Of these, two-stage least squares was by far the most widely used method in the 1960s and the early 1970s.
LISREL and PLS path modeling approaches were championed at the Cowles Commission mainly by Nobelist Trygve Haavelmo (1943), while the underlying assumptions of LISREL and PLS were challenged by statisticians such as Freedman (1987) who objected to their “failure to distinguish among causal assumptions, statistical implications, and policy claims has been one of the main reasons for the suspicion and confusion surrounding quantitative methods in the social sciences” (see also Wold’s (1987) response). Haavelmo’s path analysis never gained a large following among U.S. econometricians, but was successful in influencing a generation of Haavelmo’s fellow Scandinavian statisticians, including Hermann Wold, Karl Jöreskog , and Claes Fornell. Fornell introduced LISREL and PLS techniques to many of his Michigan colleagues through influential papers in accounting (Fornell and Larker 1981), and information systems (Davis, et al, 1989). Dhrymes (1971; Dhrymes, et al. 1974) provided evidence that PLS estimates asymptotically approached those of two-stage least squares with exactly identified equations. This point is more of academic importance than practical, because most empirical studies overidentify. But in one sense, all of the limited information methods (OLS excluded) yield similar results.
Advances in computers and the exponential increase in data storage have created many new opportunities to apply structural equation methods in the computer-intensive analysis of large datasets in complex, unstructured problems. The most popular solution techniques fall into three classes of algorithms: (1) ordinary least squares algorithms applied independently to each path, such as applied in the so-called PLS path analysis packages which may estimate with OLS or PLSR; (2) covariance analysis algorithms evolving from seminal work by Wold and his student Karl Jöreskog implemented in LISREL, AMOS, and EQS; and (3) simultaneous equations regression algorithms developed at the Cowles Commission by Tjalling Koopmans. The popularity of SEM path analysis methods in the social sciences reﬂects a more holistic, and less blatantly causal, interpretation of many real world phenomena – especially in psychology and social interaction – than may be adopted in the natural sciences. Direction in the directed network models of SEM arises from presumed cause-effect assumptions made about reality. Social interactions and artifacts are often epiphenomena – secondary phenomena that are difficult to directly link to causal factors. An example of a physiological epiphenomenon is, for example, time to complete a 100 meter sprint. I may be able to improve my sprint speed from 12 seconds to 11 seconds – but I will have difficulty attributing that improvement to any direct causal factors, like diet, attitude, weather, etc. The 1 second improvement in sprint time is an epiphenomenon – the holistic product of interaction of many individual factors.
Two main components of models are distinguished in SEM: the structural model showing potential causal dependencies between endogenous and exogenous variables, and the measurement model showing the relations between latent variables and their indicators. Exploratory and Confirmatory factor analysis models, for example, contain only the measurement part, while path diagrams can be viewed as SEMs that contain only the structural part.
In specifying pathways in a model, the modeler can posit two types of relationships: (1) free pathways, in which hypothesized causal (in fact counterfactual) relationships between variables are tested, and therefore are left 'free' to vary, and (2) relationships between variables that already have an estimated relationship, usually based on previous studies, which are 'fixed' in the model.
A modeler will often specify a set of theoretically plausible models in order to assess whether the model proposed is the best of the set of possible models. Not only must the modeler account for the theoretical reasons for building the model as it is, but the modeler must also take into account the number of data points and the number of parameters that the model must estimate to identify the model. An identified model is a model where a specific parameter value uniquely identifies the model, and no other equivalent formulation can be given by a different parameter value. A data point is a variable with observed scores, like a variable containing the scores on a question or the number of times respondents buy a car. The parameter is the value of interest, which might be a regression coefficient between the exogenous and the endogenous variable or the factor loading (regression coefficient between an indicator and its factor). If there are fewer data points than the number of estimated parameters, the resulting model is "unidentified", since there are too few reference points to account for all the variance in the model. The solution is to constrain one of the paths to zero, which means that it is no longer part of the model.
Estimation of free parameters
Parameter estimation is done by comparing the actual covariance matrices representing the relationships between variables and the estimated covariance matrices of the best fitting model. This is obtained through numerical maximization of a fit criterion as provided by maximum likelihood estimation, quasi-maximum likelihood estimation, weighted least squares or asymptotically distribution-free methods. This is often accomplished by using a specialized SEM analysis program, of which several exist.
Assessment of model and model fit
Having estimated a model, analysts will want to interpret the model. Estimated paths may be tabulated and/or presented graphically as a path model. The impact of variables is assessed using path tracing rules (see path analysis).
It is important to examine the "fit" of an estimated model to determine how well it models the data. This is a basic task in SEM modeling: forming the basis for accepting or rejecting models and, more usually, accepting one competing model over another. The output of SEM programs includes matrices of the estimated relationships between variables in the model. Assessment of fit essentially calculates how similar the predicted data are to matrices containing the relationships in the actual data.
Formal statistical tests and fit indices have been developed for these purposes. Individual parameters of the model can also be examined within the estimated model in order to see how well the proposed model fits the driving theory. Most, though not all, estimation methods make such tests of the model possible.
Of course as in all statistical hypothesis tests, SEM model tests are based on the assumption that the correct and complete relevant data have been modeled. In the SEM literature, discussion of fit has led to a variety of different recommendations on the precise application of the various fit indices and hypothesis tests.
There are differing approaches to assessing fit. Traditional approaches to modeling start from a null hypothesis, rewarding more parsimonious models (i.e. those with fewer free parameters), to others such as AIC that focus on how little the fitted values deviate from a saturated model (i.e. how well they reproduce the measured values), taking into account the number of free parameters used. Because different measures of fit capture different elements of the fit of the model, it is appropriate to report a selection of different fit measures.
Some of the more commonly used measures of fit include:
- Chi-Squared A fundamental measure of fit used in the calculation of many other fit measures. Conceptually it is a function of the sample size and the difference between the observed covariance matrix and the model covariance matrix.
- Akaike information criterion (AIC)
- Root Mean Square Error of Approximation (RMSEA)
- Another test of model fit. RMSEA values <.05 are considered to indicated good fit. An RMSEA of .1 or more is often taken to indicate poor fit.
- Standardized Root Mean Residual (SRMR)
- The SRMR is a popular absolute fit indicator. A good model should have an SRMR smaller than .05.
- Comparative Fit Index (CFI)
- In examining baseline comparisons, the CFI depends in large part on the average size of the correlations in the data. If the average correlation between variables is not high, then the CFI will not be very high. A CFI value of .90 or higher is desirable.
For each measure of fit, a decision as to what represents a good-enough fit between the model and the data must reflect other contextual factors such as sample size (for instance very large samples make the Chi-squared test overly sensitive), the ratio of indicators to factors, and the overall complexity of the model.
The model may need to be modified in order to improve the fit, thereby estimating the most likely relationships between variables. Many programs provide modification indices which may guide minor modifications. Modification indices report the change in χ² that result from freeing fixed parameters: usually, therefore adding a path to a model which is currently set to zero. Modifications that improve model fit may be flagged as potential changes that can be made to the model. Modifications to a model, especially the structural model, are changes to the theory claimed to be true. Modifications therefore must make sense in terms of the theory being tested, or be acknowledged as limitations of that theory. Changes to measurement model are effectively claims that the items/data are impure indicators of the latent variables specified by theory.
Models should not be led by MI, as Maccallum (1986) demonstrated: "even under favorable conditions, models arising from specification searches must be viewed with caution."
Sample size and power
Sample size requirements to achieve a particular significance an power in SEM hypothesis testing are similar for the same model when any of the three algorithms (PLS-PA, LISREL or systems of regression equations) are used for testing. Sample size formulas were developed in 
Interpretation and communication
The set of models are then interpreted so that claims about the constructs can be made, based on the best fitting model.
Caution should always be taken when making claims of causality even when experimentation or time-ordered studies have been done. The term causal model must be understood to mean: "a model that conveys causal assumptions," not necessarily a model that produces validated causal conclusions. Collecting data at multiple time points and using an experimental or quasi-experimental design can help rule out certain rival hypotheses but even a randomized experiment cannot rule out all such threats to causal inference. Good fit by a model consistent with one causal hypothesis invariably entails equally good fit by another model consistent with an opposing causal hypothesis. No research design, no matter how clever, can help distinguish such rival hypotheses, save for interventional experiments.
As in any science, subsequent replication and perhaps modification will proceed from the initial finding.
- Measurement invariance
- Multiple group modelling: This is a technique allowing joint estimation of multiple models, each with different sub-groups. Applications include behavior genetics, and analysis of differences between groups (e.g., gender, cultures, test forms written in different languages, etc.).
- Latent growth modeling
- Hierarchical/multilevel models; item response theory models
- Mixture model (latent class) SEM
- Alternative estimation and testing techniques
- Robust inference
- Survey sampling analyses
- Multi-method multi-trait models
- Structural Equation Model Trees
- Open source software
- Commercial packages
- Loehlin, J. C. (2004). Latent Variable Models: An Introduction to Factor, Path, and Structural Equation Analysis. Psychology Press.
- MacCallum, R. (1986). Specification searches in covariance structure modeling. Psychological Bulletin, 100, 107-120. doi
- Westland, J. Christopher (2010). "Lower bounds on sample size in structural equation modeling". Electron. Comm. Res. Appl. 9 (6): 476–487. doi:10.1016/j.elerap.2010.07.003.
- Pearl, Judea (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press. ISBN 0-521-77362-8.
- Bagozzi, R.; Yi, Y. (2012) "Specification, evaluation, and interpretation of structural equation models". Journal of the Academy of Marketing Science, 40 (1), 8–34. doi:10.1007/s11747-011-0278-x
- Bartholomew, D J, and Knott, M (1999) Latent Variable Models and Factor Analysis Kendall's Library of Statistics, vol. 7. Arnold publishers, ISBN 0-340-69243-X
- Bentler, P.M. & Bonett, D.G. (1980). "Significance tests and goodness of fit in the analysis of covariance structures". Psychological Bulletin, 88, 588-606.
- Bollen, K A (1989). Structural Equations with Latent Variables. Wiley, ISBN 0-471-01171-1
- Byrne, B. M. (2001) Structural Equation Modeling with AMOS - Basic Concepts, Applications, and Programming.LEA, ISBN 0-8058-4104-0
- Goldberger, A. S. (1972). Structural equation models in the social sciences. Econometrica 40, 979- 1001.
- Haavelmo, T. (1943) "The statistical implications of a system of simultaneous equations," Econometrica 11:1–2. Reprinted in D.F. Hendry and M.S. Morgan (Eds.), The Foundations of Econometric Analysis, Cambridge University Press, 477—490, 1995.
- Hair, J.F.; Hult, G.T.M.; Ringle, C.M.; Sarstedt, M. (2014). A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM). Thousand Oaks, CA: Sage. ISBN 9781452217444.
- Hoyle, R H (ed) (1995) Structural Equation Modeling: Concepts, Issues, and Applications. SAGE, ISBN 0-8039-5318-6
- Kaplan, D (2000) Structural Equation Modeling: Foundations and Extensions. SAGE, Advanced Quantitative Techniques in the Social Sciences series, vol. 10, ISBN 0-7619-1407-2
- Kline, R. B. (2010) Principles and Practice of Structural Equation Modeling (3rd Edition). The Guilford Press, ISBN 978-1-60623-877-6
- Jöreskog, K.; F. Yang (1996). "Non-linear structural equation models: The Kenny-Judd model with interaction effects". In G. Marcoulides and R. Schumacker, (eds.), Advanced structural equation modeling: Concepts, issues, and applications. Thousand Oaks, CA: Sage Publications.
- Ed Rigdon's Structural Equation Modeling Page: people, software and sites
- Structural equation modeling page under David Garson's StatNotes, NCSU
- Issues and Opinion on Structural Equation Modeling, SEM in IS Research
- The causal interpretation of structural equations (or SEM survival kit) by Judea Pearl 2000.
- Structural Equation Modeling Reference List by Jason Newsom: journal articles and book chapters on structural equation models
- PLS-SEM book: online resources and additional information
- Path Analysis in AFNI: The open source (GPL) AFNI package contains SEM code
- Handbook of Management Scales, a collection of previously used multi-item scales to measure constructs for SEM